First, observe that
\(\begin{array}{ccl}\gamma (t)&=&\displaystyle \sum \limits _{r>s}x_{rs}(t)(d\phi _{\exp (tw_{ij})})_o(\operatorname{\mathrm {Ad}}(\exp (-tw_{ij}))w_{rs})^*(o)\\&=&\displaystyle \sum \limits _{r>s}x_{rs}(t)(d\phi _{\exp (tw_{ij})})_o(e^{-t\cdot \operatorname{\mathrm {ad}}(w_{ij})}w_{rs})^*(o),\end{array}\)
and
\(\displaystyle \left.\frac{d}{dt}\exp (tw_{ij})K_{\emptyset }=\frac{d}{ds}\exp ((t+s)w_{ij})K_{\emptyset }\ \right|_{s=0}=\left.\frac{d}{ds}\exp (tw_{ij})\exp (sw_{ij})K_{\emptyset }\ \right|_{s=0}=(d\phi _{\exp (tw_{ij})})_ow_{ij}^*(o),\)
thus
\(\begin{array}{ccl}\displaystyle \nabla _{\dot{\zeta }}\gamma &=&\displaystyle \nabla _{\dot{\zeta }}\left(\sum \limits _{r>s}x_{rs}(t)(d\phi _{\exp (tw_{ij})})_o(e^{-t\cdot \operatorname{\mathrm {ad}}(w_{ij})}w_{rs})^*(o)\right)\\\\&=&\displaystyle \sum \limits _{r>s}x^{\prime }_{rs}(t)(d\phi _{\exp (tw_{ij})})_o(e^{-t\cdot \operatorname{\mathrm {ad}}(w_{ij})}w_{rs})^*(o)+\sum \limits _{r>s}x_{rs}(t)\nabla _{\dot{\zeta }}(d\phi _{\exp (tw_{ij})})_o(e^{-t\cdot \operatorname{\mathrm {ad}}(w_{ij})}w_{rs})^*(o)\\\\&=&\displaystyle \sum \limits _{r>s}x^{\prime }_{rs}(t)(d\phi _{\exp (tw_{ij})})_o(e^{-t\cdot \operatorname{\mathrm {ad}}(w_{ij})}w_{rs})^*(o)\\\\&&+\displaystyle \sum \limits _{r>s}x_{rs}(t)\nabla _{(d\phi _{\exp (tw_{ij})})_ow_{ij}^*}(d\phi _{\exp (tw_{ij})})_o(e^{-t\cdot \operatorname{\mathrm {ad}}(w_{ij})}w_{rs})^*(o)\\\\&=&\displaystyle (d\phi _{\exp (tw_{ij})})_o\left(\sum \limits _{r>s}x^{\prime }_{rs}(t)(e^{-t\cdot \operatorname{\mathrm {ad}}(w_{ij})}w_{rs})^*(o)+\sum \limits _{r>s}x_{rs}(t)\nabla _{w_{ij}^*}(e^{-t\cdot \operatorname{\mathrm {ad}}(w_{ij})}w_{rs})^*(o)\right).\end{array}\)
Therefore, \(\gamma \)  is parallel along \(\exp (tw_{ij})K_{\emptyset }\)  if and only if
\((d\phi _{\exp (tw_{ij})})_o^{-1}(\nabla _{\dot{\zeta }}\gamma )=\sum \limits _{r>s}x^{\prime }_{rs}(t)(e^{-t\cdot \operatorname{\mathrm {ad}}(w_{ij})}w_{rs})^*(o)+\sum \limits _{r>s}x_{rs}(t)\nabla _{w_{ij}^*}(e^{-t\cdot \operatorname{\mathrm {ad}}(w_{ij})}w_{rs})^*(o)=0.\) 
We recall that for \(X,Y\in \mathfrak {so}(n)\) , \(\nabla _{X^*}Y^*(o)=(-\frac{1}{2}[X,Y]+U(X,Y))^*(o)\)  ([1]}), where the operator \(U:\mathfrak {so}(n)\times \mathfrak {so}(n)\rightarrow \mathfrak {so}(n)\)  is given by the equality
\(2g(U(X,Y),Z)=g([Z,X],Y)+g([Z,Y],X),\ Z\in \mathfrak {so}(n).\) 
By using the relations
\(&e^{-t\cdot \operatorname{\mathrm {ad}}(w_{ij})}w_{rs}=w_{rs} \text{ if } (r,s)=(i,j) \text{ or } r,s\notin \lbrace i,j\rbrace ,\\&e^{-t\cdot \operatorname{\mathrm {ad}}(w_{ij})}w_{is}=\cos (t)w_{is}+\sin (t)w_{js},\ s\ne j,\ s>i,\\&e^{-t\cdot \operatorname{\mathrm {ad}}(w_{ij})}w_{js}=\cos (t)w_{js}-\sin (t)w_{is},\ s<j,\\&e^{-t\cdot \operatorname{\mathrm {ad}}(w_{ij})}w_{si}=\cos (t)w_{si}+\sin (t)w_{sj},\ s>i,\\&e^{-t\cdot \operatorname{\mathrm {ad}}(w_{ij})}w_{sj}=\cos (t)w_{sj}-\sin (t)w_{si},\ s\ne i,\ s>j,\\&U(w_{ij},w_{rs})=0 \text{ if } (r,s)=(i,j) \text{ or } r,s\notin \lbrace i,j\rbrace \\&U(w_{ij},w_{is})=\displaystyle \frac{\mu _{ij}-\mu _{is}}{2}w_{js},\ s\ne j,\ s>i,\\&U(w_{ij},w_{js})=\displaystyle \frac{\mu _{js}-\mu _{ij}}{2}w_{is},\ s<j,\\&U(w_{ij},w_{si})=\displaystyle \frac{\mu _{ij}-\mu _{si}}{2}w_{sj},\ s>i,\\&U(w_{ij},w_{sj})=\displaystyle \frac{\mu _{sj}-\mu _{ij}}{2}w_{si},\ s\ne i,\ s>j,\\\) 

we have that \((d\phi _{\exp (tw_{ij})})_o^{-1}(\nabla _{\dot{\zeta }}\gamma )\)  is equal to
\(&\displaystyle x^{\prime }_{ij}(t)w_{ij}^*(o)+\sum \limits _{{{\scriptstyle \begin{matrix}r>s\\r,s\notin \lbrace i,j\rbrace \end{matrix}}}}x^{\prime }_{rs}(t)w_{rs}^*(o)\\&+\sum \limits _{{{\scriptstyle \begin{matrix}s<i\\s\ne j\end{matrix}}}}x^{\prime }_{is}(t)(\cos (t)w_{is}^*+\sin (t)w_{js}^*)(o)+\sum \limits _{s<j}x^{\prime }_{js}(t)(\cos (t)w_{js}^*-\sin (t)w_{is}^*)(o)\\&+\sum \limits _{s>i}x^{\prime }_{si}(t)(\cos (t)w_{si}^*+\sin (t)w_{sj}^*)(o)+\sum \limits _{{{\scriptstyle \begin{matrix}s>j\\s\ne i\end{matrix}}}}x^{\prime }_{sj}(t)(\cos (t)w_{sj}^*-\sin (t)w_{si}^*)(o)\\&+\sum \limits _{{{\scriptstyle \begin{matrix}s<i\\s\ne j\end{matrix}}}}\left\lbrace x_{is}(t)(-\sin (t))w_{is}^*(o)+x_{is}(t)\cos (t)\left(-\frac{1}{2}[w_{ij},w_{is}]+U(w_{ij},w_{is})\right)^*(o)\right\rbrace \\&+\sum \limits _{{{\scriptstyle \begin{matrix}s<i\\s\ne j\end{matrix}}}}\left\lbrace x_{is}(t)\cos (t)w_{js}^*(o)+x_{is}(t)\sin (t)\left(-\frac{1}{2}[w_{ij},w_{js}]+U(w_{ij},w_{js})\right)^*(o)\right\rbrace \\&+\sum \limits _{s<j}\left\lbrace x_{js}(t)(-\sin (t))w_{js}^*(o)+x_{js}(t)\cos (t)\left(-\frac{1}{2}[w_{ij},w_{js}]+U(w_{ij},w_{js})\right)^*(o)\right\rbrace \\&-\sum \limits _{s<j}\left\lbrace x_{js}(t)\cos (t)w_{is}^*(o)+x_{js}(t)\sin (t)\left(-\frac{1}{2}[w_{ij},w_{is}]+U(w_{ij},w_{is})\right)^*(o)\right\rbrace \\&+\sum \limits _{s>i}\left\lbrace x_{si}(t)(-\sin (t))w_{si}^*(o)+x_{si}(t)\cos (t)\left(-\frac{1}{2}[w_{ij},w_{si}]+U(w_{ij},w_{si})\right)^*(o)\right\rbrace \\&+\sum \limits _{s<i}\left\lbrace x_{si}(t)\cos (t)w_{sj}^*(o)+x_{si}(t)\sin (t)\left(-\frac{1}{2}[w_{ij},w_{sj}]+U(w_{ij},w_{sj})\right)^*(o)\right\rbrace \\&+\sum \limits _{{{\scriptstyle \begin{matrix}s>j\\s\ne i\end{matrix}}}}\left\lbrace x_{sj}(t)(-\sin (t))w_{sj}^*(o)+x_{sj}(t)\cos (t)\left(-\frac{1}{2}[w_{ij},w_{sj}]+U(w_{ij},w_{sj})\right)^*(o)\right\rbrace \\&-\sum \limits _{{{\scriptstyle \begin{matrix}s>j\\s\ne i\end{matrix}}}}\left\lbrace x_{sj}(t)\cos (t)w_{si}^*(o)+x_{sj}(t)\sin (t)\left(-\frac{1}{2}[w_{ij},w_{si}]+U(w_{ij},w_{si})\right)^*(o)\right\rbrace \\&=\displaystyle x^{\prime }_{ij}(t)w_{ij}^*(o)+\sum \limits _{{{\scriptstyle \begin{matrix}r>s\\r,s\notin \lbrace i,j\rbrace \end{matrix}}}}x^{\prime }_{rs}(t)w_{rs}^*(o)\\&+\sum \limits _{j<s<i}x^{\prime }_{is}(t)\cos (t)w_{is}^*(o)+\sum \limits _{s<j}x^{\prime }_{is}(t)\cos (t)w_{is}^*(o)-\sum \limits _{j<s<i}x^{\prime }_{is}(t)\sin (t)w_{sj}^*(o)\\&+\sum \limits _{s<j}x^{\prime }_{is}(t)\sin (t)w_{js}^*(o)+\sum \limits _{s<j}x^{\prime }_{js}(t)\cos (t)w_{js}^*(o)-\sum \limits _{s<j}x^{\prime }_{js}(t)\sin (t)w_{is}^*(o)\\&+\sum \limits _{s>i}x^{\prime }_{si}(t)\cos (t)w_{si}^*(o)+\sum \limits _{s>i}x^{\prime }_{si}(t)\sin (t)w_{sj}^*(o)+\sum \limits _{s>i}x^{\prime }_{sj}(t)\cos (t)w_{sj}^*(o)\\&+\sum \limits _{j<s<i}x^{\prime }_{sj}(t)\cos (t)w_{sj}^*(o)-\sum \limits _{s>i}x^{\prime }_{sj}(t)\sin (t)w_{si}^*(o)+\sum \limits _{j<s<i}x^{\prime }_{sj}(t)\sin (t)w_{is}^*(o)\\&-\sum \limits _{j<s<i}x_{is}(t)\sin (t)w_{is}^*(o)-\sum \limits _{s<j}x_{is}(t)\sin (t)w_{is}^*(o)-\sum \limits _{j<s<i}x_{is}(t)\cos (t)\left(\frac{1+\mu _{ij}-\mu _{is}}{2}\right)w_{sj}^*(o)\\&+\sum \limits _{s<j}x_{is}(t)\cos (t)\left(\frac{1+\mu _{ij}-\mu _{is}}{2}\right)w_{js}^*(o)-\sum \limits _{j<s<i}x_{is}(t)\cos (t)w_{sj}^*(o)+\sum \limits _{s<j}x_{is}(t)\cos (t)w_{js}^*(o)\\&-\sum \limits _{j<s<i}x_{is}(t)\sin (t)\left(\frac{1+\mu _{ij}-\mu _{sj}}{2}\right)w_{is}^*(o)-\sum \limits _{s<j}x_{is}(t)\sin (t)\left(\frac{1+\mu _{ij}-\mu _{js}}{2}\right)w_{is}^*(o)\\&-\sum \limits _{s<j}x_{js}(t)\sin (t)w_{js}^*(o)-\sum \limits _{s<j}x_{js}(t)\cos (t)\left(\frac{1+\mu _{ij}-\mu _{js}}{2}\right)w_{is}^*(o)-\sum \limits _{s<j}x_{js}(t)\cos (t)w_{is}^*(o)\\&-\sum \limits _{s<j}x_{js}(t)\sin (t)\left(\frac{1+\mu _{ij}-\mu _{is}}{2}\right)w_{js}^*(o)-\sum \limits _{s>i}x_{si}(t)w_{si}^*(o)+\sum \limits _{s>i}x_{si}(t)\cos (t)\left(\frac{1+\mu _{ij}-\mu _{si}}{2}\right)w_{sj}^*(o)\\&+\sum \limits _{s>i}x_{si}(t)\cos (t)w_{sj}^*(o)-\sum \limits _{s>i}x_{si}(t)\sin (t)\left(\frac{1+\mu _{ij}-\mu _{sj}}{2}\right)w_{si}^*(o)-\sum \limits _{j<s<i}x_{sj}(t)\sin (t)w_{sj}^*(o)\\&-\sum \limits _{s>i}x_{sj}(t)\sin (t)w_{sj}^*(o)-\sum \limits _{s>i}x_{sj}(t)\cos (t)\left(\frac{1+\mu _{ij}-\mu _{sj}}{2}\right)w_{si}^*(o)\\&+\sum \limits _{j<s<i}x_{sj}(t)\cos (t)\left(\frac{1+\mu _{ij}-\mu _{sj}}{2}\right)w_{is}^*(o)+\sum \limits _{j<s<i}x_{sj}(t)\cos (t)w_{is}^*(o)-\sum \limits _{s>i}x_{sj}(t)\cos (t)w_{si}^*(o)\\&-\sum \limits _{s>i}x_{sj}(t)\sin (t)\left(\frac{1+\mu _{ij}-\mu _{si}}{2}\right)w_{sj}^*(o)-\sum \limits _{j<s<i}x_{sj}(t)\sin (t)\left(\frac{1+\mu _{ij}-\mu _{is}}{2}\right)w_{sj}^*(o)\\&=\displaystyle x^{\prime }_{ij}(t)w_{ij}^*(o)+\sum \limits _{{{\scriptstyle \begin{matrix}r>s\\r,s\notin \lbrace i,j\rbrace \end{matrix}}}}x^{\prime }_{rs}(t)w_{rs}^*(o)\\&+\sum \limits _{j<s<i}\left\lbrace x^{\prime }_{is}(t)\cos (t)+x^{\prime }_{sj}(t)\sin (t)+\left(\frac{3+\mu _{ij}-\mu _{sj}}{2}\right)(-x_{is}(t)\sin (t)+x_{sj}(t)\cos (t))\right\rbrace w_{is}^*(o)\\&+\sum \limits _{j<s<i}\left\lbrace -x^{\prime }_{is}(t)\sin (t)+x^{\prime }_{sj}(t)\cos (t)+\left(\frac{3+\mu _{ij}-\mu _{is}}{2}\right)(-x_{is}(t)\cos (t)-x_{sj}(t)\sin (t))\right\rbrace w_{sj}^*(o)\\&+\sum \limits _{s<j}\left\lbrace x^{\prime }_{is}(t)\cos (t)-x^{\prime }_{js}(t)\sin (t)+\left(\frac{3+\mu _{ij}-\mu _{js}}{2}\right)(-x_{is}(t)\sin (t)-x_{js}(t)\cos (t))\right\rbrace w_{is}^*(o)\\&+\sum \limits _{s<j}\left\lbrace x^{\prime }_{is}(t)\sin (t)+x^{\prime }_{js}(t)\cos (t)+\left(\frac{3+\mu _{ij}-\mu _{is}}{2}\right)(x_{is}(t)\cos (t)-x_{js}(t)\sin (t))\right\rbrace w_{js}^*(o)\\&+\sum \limits _{s>i}\left\lbrace x^{\prime }_{si}(t)\cos (t)-x^{\prime }_{sj}(t)\sin (t)+\left(\frac{3+\mu _{ij}-\mu _{sj}}{2}\right)(-x_{si}(t)\sin (t)-x_{sj}(t)\cos (t))\right\rbrace w_{si}^*(o)\\&+\sum \limits _{s>i}\left\lbrace x^{\prime }_{si}(t)\sin (t)+x^{\prime }_{sj}(t)\cos (t)+\left(\frac{3+\mu _{ij}-\mu _{si}}{2}\right)(x_{si}(t)\cos (t)-x_{sj}(t)\sin (t))\right\rbrace w_{sj}^*(o),\) 

so we have the result.
